Synthetic Division 1 March 2011

Synthetic Division A trick for dividing polynomials Helps us solve for the roots of polynomials Only works when we divide by 1st degree (linear) polynomials My degree can’t be larger than 1!

Synthetic Division

Your Turn On the Synthetic Division – Guided Notes handout, complete problems 1 – 5. You will: Decide if it’s possible to use synthetic division to divide the two polynomials

Division Vocab Review Dividend Divisor Quotient

Preparing for Synthetic Division Can only be used when the divisor is in the form If the divisor isn’t in the form x – c, then you must convert the expression to include subtraction. x – c

Preparing for Synthetic Division, cont.

Preparing for Synthetic Division, cont. Polynomials need to be written in expanded, standard polynomial form. Translation: If you’re missing terms, then you need to write them out as 0 times (*) the variable.

Preparing for Synthetic Division, cont.

Your Turn On Synthetic Division - Guided Notes handout, write the dividend in expanded standard polynomial form for problems 6 – 10. Write the divisor in the form x – c.

*Synthetic Division Steps Example Problem:

Prep Step Divisor x – c? x – 2 Dividend in Expanded Standard Polynomial Form? 3x4 – 8x2 – 11x + 1 3x4 + – 8x2 – 11x + 1 3x4 + 0x3 – 8x2 – 11x + 1

Step 1 2 Write the constant value of the divisor (c) here.

Step 2 2 3 0 -8 -11 1 Write all the coefficients of the expanded dividend here.

Step 3 2 3 0 -8 -11 1 3 “Drop” the 1st coefficient underneath the line.

Step 4 2 3 0 -8 -11 1 6 3 Multiply “c” by the last value underneath the line. Write their product just underneath the next coefficient.

Step 5 2 3 0 -8 -11 1 6 3 6 Add together the numbers in that column and write their sum underneath the line.

Step 6 2 3 0 -8 -11 1 6 12 3 6 Multiply “c” by the last value underneath the line. Write their product just underneath the next coefficient.

Step 7 2 3 0 -8 -11 1 6 12 8 -6 3 6 4 -3 -5 Repeat steps 5 and 6 until a number appears in the box underneath the last column.

Step 8 – Naming the Quotient 2 3 0 -8 -11 1 6 12 8 -6 3 6 4 -3 -5 In the last row are the coefficients of the quotient in decreasing order. The quotient is one degree less than the dividend.

Step 8 – Naming the Quotient 3 6 4 -3 -5 The number in the box is the remainder. 3x3 + 6x2 + 4x – 3 Remainder -5

Your Turn On the Synthetic Division – Guided Notes handout, solve for the quotient of problems 11 – 14 using synthetic division

Synthetic Division and the Factor Theorem Conclusions:

Your Turn: Using problems 1 – 12 on the Synthetic Division Practice handout (last night’s hmwk), identify which problems represent division by a factor/root and which problems represent division by NOT a factor root.

So What’s Next? * To get the remaining roots, set the expression equal to 0, factor, and solve.

Your Turn: On the Synthetic Division Practice handout, solve for the remaining roots for problems 1 – 4 and 10 – 12

Rewriting the Original Polynomial We can use the roots and linear factors to rewrite the polynomial This form is called the product of linear factors If you multiplied all the linear factors together, then you’d get the original polynomial

Reminder: Roots vs. Linear Factors

Product of Linear Factors Product = Multiply Product of linear factors = Multiply all the linear factors Translation: Rewrite all the linear factors with parentheses around each factor Helpful format for graphing polynomials Product of Linear Factors